Magnetic Fields

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Magnetic Fields (Lorentz Force and Magnetic
Force)
Lectures 16, 17, 18
Ph 2B Lectures by George M. Fuller, UCSD
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George M. Fuller 427 SERF gfuller_at_ucsd.edu 822-1
214
Office Hours 200 PM - 330 PM Tuesday 329 SERF
PM
TH 700 PM - 800 PM WLH 2204
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no quiz
Ch 29,30,31
Ch 31,32
Ch 33
problem assignment TBA
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The origin of magnetic force and magnetic
phenomena is moving electric charge (currents).
We can summarize our picture for the physics of
magnetism as follows (1) Moving electric
charges (currents) generate magnetic fields. (2)
When other moving charges encounter these fields,
they feel a force.
We will follow our book, Wolfson Pasachoff, and
first discuss (2) and then later move on to (1).
We will eventually see that electric and magnetic
forces on charges are different manifestations of
the same underlying phenomenon the
electromagnetic interaction.
In much of the following development it will be
important for you to understand the vector cross
product. To review this look at Chapter 13 on
rotation.
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Lorentz Force Law
We can define the magnetic field B (a vector
quantity) at a point by the vector force Fmag at
that point experienced by a particle with charge
q and velocity v
If there is also an electric field E at this
point, then in addition to the above magnetic
force, there will be an electric force
FelecqE and the total force Ftot on the charge
will be
Ph 2B Lectures by George M. Fuller, UCSD
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The (vector) magnetic force on the charge q at a
particular point in space depends on the (vector)
velocity of the charge and on the (vector)
magnetic field at this point
The magnitude of this force is
Ph 2B Lectures by George M. Fuller, UCSD
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What about the direction of the force?
Since the force is given by the vector cross
product of velocity and magnetic field, it is
orthogonal to both of these. That is, the force
vector will be perpendicular to the plane defined
by the velocity and magnetic field vectors.
Fmag
B
q
v
Ph 2B Lectures by George M. Fuller, UCSD
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Some other properties of the vector cross product
It is not commutative
When working with vector components, it is
sometimes useful to remember the cross products
of the unit vectors that point along the
coordinate axes
The BAC CAB rule
Ph 2B Lectures by George M. Fuller, UCSD
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Dimensions of Magnetic Field
Recall the dimensions of electric field
Ph 2B Lectures by George M. Fuller, UCSD
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Magnetic Field magnitudes in the lab and in
nature
Ph 2B Lectures by George M. Fuller, UCSD
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Example Problem 9 An alpha particle (2
protons, 2 neutrons) is moving
with velocity v (given below) in a magnetic field
B (given below). Find the
magnetic force on the particle.
First note that the charge on this particle is
The force is a vector given by
Note the dimensions in each of these products
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Example Problem 9 An alpha particle (2
protons, 2 neutrons) is moving
with velocity v (given below) in a magnetic field
B (given below). Find the
magnetic force on the particle. Continued . . .
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The motion of a charged particle in a magnetic
field
Note first that the magnetic (Lorentz) force can
never do work on a charged particle. This is
because the magnetic force is always at right
angles to the charged particles velocity vector.
Therefore, the magnetic force by itself can never
change the kinetic energy or the speed of a
charged particle.
Ph 2B Lectures by George M. Fuller, UCSD
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Consider the motion of a a charged particle in a
uniform magnetic field.
Let us take the case where the particles
velocity is in the plane of the screen and the
uniform magnetic field points out of the screen
Result uniform circular motion
B-field (tips of arrows)
Fmag
v
Fmag
v
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In this case, we have uniform circular motion
with the centripetal acceleration supplied by the
magnetic force
where r is the radius of the circular path
v
Fmag
r
Ph 2B Lectures by George M. Fuller, UCSD
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Example Problem 27 (see the figure on the
overhead) As shown, ions are accelerated through
a potential difference V after which they enter a
region with a uniform magnetic field and then
follow semicircular paths to a detector at x from
where leave the accelerator and enter the
non-zero B-field region. Show how x depends on
the charge to mass ratio (q/m) of a particle.
The speed of the particles when they enter the
B-field region can be obtained by equating the
kinetic energy to the potential energy drop in
the accelerator
Particles are subsequently bent into a
semicircular path with diameter 2r. Note x 2r.
By measuring the number of ions that pile up at
different positions x we find the
relative abundances of various atomic or
molecular species in a sample. The ions are
sorted by their charge to mass ratios. This is
the Mass Spectrometer.
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Note that, in these examples, a velocity
component for the particle that lies along the
B-field lines (in the direction of the B-field,
or opposite to it) is unaffected by the magnetic
force from a uniform B-field. Therefore,
particles will tend to spiral along magnetic
field lines. There are many examples of this in
nature. In fact, in regions of space where
B-field lines converge (regions of increasing
magnetic field strength) particles can be slowed
down, stopped and even have their spiraling
motion reversed. This magnetic trapping effect is
exploited in the design of magnetic bottles and
is seen in, for example, the behavior of charged
particles in the aurora.
B field lines
Fmag
Ph 2B Lectures by George M. Fuller, UCSD
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The Cyclotron Frequency
The period of the circular orbit of a charged
particle in a uniform magnetic field is
Note that this is independent of the particles
velocity !
The relation between the period and the frequency
of the motion is
The cyclotron frequency depends only on the
magnetic field strength and on the particles
charge to mass ratio.
Charged particles emit electromagnetic radiation
at the cyclotron frequency.
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The Magnetic Force on a Current
The force on a segment of straight wire with
length l can be obtained by realizing that each
charge carrier in the wire has a drift velocity
vd and charge q. Therefore, the force on each
charge is
For a number density n of charge carriers, the
current density (flux of charge) is
If the wire has cross sectional area A then a
current I flows, where
For a straight, uniform segment of the wire of
length l there will be n A l charges and, as a
result, the total magnetic force on the segment
will be
Comparing to the above, we conclude that
the force on the segment is
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We can easily generalize this result to a
configuration where the conductor carrying the
current is not straight and where the magnetic
field is not uniform
First consider the magnetic force on an
infinitesimal length of the wire
The total force is just the sum along the
conductor
current I
B-field
dl
Where in this expression we envision integrating
along the conductor (wire) and we note that all
quantities, (vector) magnetic field, (vector)
infinitesimal line segment length, and current
could depend on position and must therefore, in
general, be under the integral sign.
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Example Problem 41 A wire carrying a 1.5 A
current passes through a region containing a 48
m-T magnetic field (directed into the screen).
The wire is perpendicular to the field and makes
a quarter circle turn of radius 21 cm as it
passes through the field region. Find the
magnitude and direction of the force on this
section of wire.
uniform B-field, 48 m-T, into screen
y axis
lay down x and y axes as shown (z axis is
out of screen)
I 1.5 A
x axis
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Current Loop in a Magnetic Field
A current loop is a closed circuit carrying
current. These configurations of current give
rise to the familiar dipole magnetic field line
arrangements seen in objects as diverse as
refrigerator magnets, planets, stars, and
neutrons.
Consider a rectangular current loop in a uniform
magnetic field, with the normal to the plane of
the loop making an angle q with the magnetic
field
B
Seen edge on (from the top)
q
F
I
B
a
I
q
q
I
A
F
b
q
Top and bottom forces on the loop cancel, but
forces on the sides give a torque. For one side
we have. . .
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A small current loop (of any shape) constitutes a
magnetic dipole and when placed in a magnetic
field feels a torque given by the formula we just
derived, which we can re-write as
Where we have defined the magnetic dipole moment
Curl the fingers of your right hand in the
direction of the current in the loop and your
thumb will point in the direction of the magnetic
dipole moment.
If there are N turns or loops of wire around
the area of the loop then the magnetic dipole
moment is increased in magnitude
It is clear that it takes work to twist the
magnetic dipole moment out of alignment with the
B-field, suggesting that we can define
a potential energy
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Example Problem 72 A square wire loop of mass m
carries a current I. It is initially
in equilibrium, with its magnetic moment vector
aligned with a uniform magnetic field B. The loop
is rotated slightly out of equilibrium about an
axis through the centers of the two opposite
sides and then released. Show that it executes
simple harmonic motion and find the period of
oscillation.
Solution Note that when the normal to the loop
area is displaced by angle q from the magnetic
field direction the resulting torque is in the
direction to restore equilibrium. Setting torque
equal to the product of moment of inertia and
angular acceleration, we get
See table 12.2 pg. 296 and use the parallel axis
theorem.
The moment of inertia of the loop is given by
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Charge as a Source of Electric Field
If there are charges (charge density r(x,y,z))
but the magnetic field is not changing in
time, it turns out that the electric field
distribution in space is related to the charge
density in space by
electric field at position (vector) r1
unit vector along direction r12 r1 -
r2. (vector that points from r2 to r1 )
charge density at position (vector) r2 r(r2)
r(x2,y2,z2)
Where Coulombs constant is
Ph 2B Lectures by George M. Fuller, UCSD
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Currents as Sources of Magnetic Field
In conditions where currents flow but where the
electric field is not changing in time, it turns
out that the magnetic field distribution in space
is related to the current density in space by
magnetic field at position (vector) r1
unit vector along direction r12 r1 -
r2. (vector that points from r2 to r1 )
current density at position (vector) r2
Define the vacuum permeability constant
Ph 2B Lectures by George M. Fuller, UCSD
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We had the following relation, which is called
the Law of Biot Savart. You can see that it is
a kind of Coulombs Law for magnetic fields.
Like Coulombs Law, it is valid only when fields
are not changing in time
Let us say that we have a thin uniform wire with
cross sectional area A and arbitrary shape and
carrying current I.
A
j
r1
r2
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Example Find the magnetic field at a distance x
from the center of a ring carrying a current I
and radius a with geometry as shown.
Assume that the ring is very thin and the point
where we want the magnetic field is on the
symmetry axis.
I
q
a
q
x
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Magnetic Flux
The Magnetic Flux FB through a surface S is
defined to be the surface integral
Where an element of surface area is a
vector whose magnitude is given by the area of
the element and whose direction is perpendicular
to the tangent plane of the surface element at
its center.
q
dx
dy
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Consider a surface S, threaded by a magnetic
field.
Tile the surface with area elements and
evaluate B dA for each and sum.
dA
B
S
The result will be the magnetic flux through the
surface
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Gausss Law for Magnetic Fields
The total flux of magnetic field through surface
R.
There are no magnetic charges that is,
no magnetic monopoles.
Ph 2B Lectures by George M. Fuller, UCSD